[span_0](start_span)[span_0](end_span)[span_1](start_span)[span_1](end_span)
Unlike an explicit rule which lets you "time travel" to any position, a recursive rule forces you to build the sequence step-by-step. It relates a term to its previous terms. [span_2](start_span)[span_2](end_span)
If you know earlier terms of the sequence, then you can find the next terms using the rule. A famous example is the VirahΔnka-Fibonacci sequence, where every term is simply the sum of the two terms before it! [span_3](start_span)[span_3](end_span)[span_4](start_span)[span_4](end_span)
If you know earlier terms of the sequence, then you can find the next terms using the rule. A famous example is the VirahΔnka-Fibonacci sequence, where every term is simply the sum of the two terms before it! [span_3](start_span)[span_3](end_span)[span_4](start_span)[span_4](end_span)
1 Evaluating Recursive Rules
2 Advanced Recursive Substitution
Select the correct option:
What is the defining characteristic of a recursive rule for a sequence? [span_13](start_span)[span_13](end_span)
Concept:
1
A recursive formula requires one or more starting values (base cases) and an equation showing how to get the next term from the preceding ones. [span_14](start_span)[span_14](end_span)
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Hint:
1
Look for the option that talks about "previous terms".
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Mistakes to Avoid:
!
Confusing with Explicit Rules: Do not select Option A. Explicit rules use position \(n\). Recursive rules use \(u_{n-1}\). [span_15](start_span)[span_15](end_span)
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Solution:
As stated in the textbook, describing a sequence by relating terms to previous terms is known as a recursive rule. [span_16](start_span)[span_16](end_span)
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Select the correct option:
In the VirahΔnka-Fibonacci sequence, how is each new term calculated? [span_17](start_span)[span_17](end_span)
Concept:
1
A recursive rule does not only have to involve the single previous term - it could involve the previous two or more terms. [span_18](start_span)[span_18](end_span)
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Hint:
1
The sequence is: 1, 2, 3, 5, 8, 13...
2
Notice that \(1+2=3\), \(2+3=5\), \(3+5=8\). [span_19](start_span)[span_19](end_span)
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Mistakes to Avoid:
!
Assuming it requires multiplication: Because it grows quickly, some assume multiplication is involved, but it is pure addition.
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Solution:
The textbook states the formula for the VirahΔnka-Fibonacci sequence is \(V_n = V_{n-1} + V_{n-2}\). [span_20](start_span)[span_20](end_span)
This explicitly means each term is obtained by adding the previous two. [span_21](start_span)[span_21](end_span)
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Select the correct option:
Who is historically credited as the first to explicitly write down and study the sequence \(1, 2, 3, 5, 8 \dots\) in the context of poetry and meter? [span_22](start_span)[span_22](end_span)
Concept:
1
Mathematical discoveries are often named after European mathematicians, despite being discovered centuries earlier by Indian mathematicians. [span_23](start_span)[span_23](end_span)
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Hint:
1
Look for the 7th century CE mathematician mentioned in the textbook's historical note. [span_24](start_span)[span_24](end_span)
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Mistakes to Avoid:
!
Assuming Fibonacci was first: Fibonacci popularized it in Europe in 1200 CE, but he was not the first to write it down explicitly! [span_25](start_span)[span_25](end_span)
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Solution:
The textbook states it was first written down explicitly and studied by VirahΔnka in his work VrttajΔtisamuchaya in the 7th century CE. [span_26](start_span)[span_26](end_span)
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Select the correct option:
If a sequence is defined by the recursive rule \(t_1 = 4\) and \(t_n = t_{n-1} + 5\), what is the value of \(t_3\)?
Concept:
1
A recursive formula requires you to find every preceding term before you can find the target term.
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Hint:
1
Find \(t_2\) first: \(t_2 = t_1 + 5\).
2
Then use \(t_2\) to find \(t_3\).
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Mistakes to Avoid:
!
Treating it explicitly: Do NOT plug \(n=3\) into \(t_n = 4 + 5(n)\). The formula is recursive, not explicit!
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Solution:
Given: \(t_1 = 4\) and \(t_n = t_{n-1} + 5\)
\(t_2 = t_1 + 5 = 4 + 5 = 9\)
\(t_3 = t_2 + 5 = 9 + 5 = 14\)
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Select the correct option:
If you are given the recursive formula \(u_n = 3u_{n-1}\) but you are not given the value of \(u_1\), can you find the 5th term?
Concept:
1
A complete recursive definition requires TWO parts: a base case (the starting term) and the recursive equation. [span_27](start_span)[span_27](end_span)
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Hint:
1
If the rule is "multiply by 3", what are you supposed to multiply by 3 to get the first step going?
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Mistakes to Avoid:
!
Assuming a starting point: Never assume the sequence starts at 0 or 1 unless it is explicitly provided in the problem statement.
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Solution:
Without the starting term (\(u_1\)), the recursive formula cannot be evaluated because there is no initial value to plug into \(u_{n-1}\). [span_28](start_span)[span_28](end_span)
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★ Complex Recursive Chains
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